Computational Complexity and other fun stuff in math and computer science from Lance Fortnow and Bill Gasarch

Tuesday, March 21, 2006

Favorite Theorems: Relativization

February Edition
After the work of Cook and Karp popularized the P versus NP question,
computer scientists immediately tried hard to prove P=NP or P≠NP. Baker,
Gill and Solovay showed that most of their approaches were doomed to
failure.

Baker, Gill and Solovay noted that complexity proofs relativized, that
is held even if all machines involved could make queries to some
"oracle," i.e., making queries to some fixed set. They
created oracles A and B such that

PA = NPA

PB ≠ NPB

If one had a relativizable proof that P ≠ NP then the proof would
also show PA ≠ NPA, contradicting their
theorem. Baker et. al. give a few more relativization results and we've seen
hundreds more since.
The paper gives very little about the
philosophical implications of their results. For a short while some researchers
thought that these results could lead to true independence of the P
versus NP question, but this thinking was quickly abandoned and later
we have seen some theorems that do not relativize, particularly in the
area of interactive proof systems.
Relativization results do help us understand what theorems to pursue,
what techniques cannot solve our questions. Nearly all the techniques
we know for time classes, outside of the algebraic techniques used
for interactive proofs, do relativize. Only a very few of the
known relativization results later had proofs in the
opposite direction.
For more read my 1994 survey The Role of Relativization in Complexity Theory.